Gargantula is a timed boss fight with approximately 12 minutes to be defeated before she feeds, at which point the event fails and she disappears.

Essentially, you have to kill Gargantula 4 times; each time you take her health down to a certain percentage (1st time 75%: 2nd time 50%: 3rd time 25%) she will regain all of her health, summon many spiderlings and become invulnerable to damage as long as any spiderlings are alive. Once all of the spiderlings are defeated, Gargantula becomes vulnerable to damage again, and with seemingly less armor each time. The third time you will actually be able to defeat her and gain the rewards.

While she doesn't hit very hard, it is easy to kite Gargantula by strafing in circles around her (even in melee).

Make sure to bring poison/condition management skills, as Gargantula and the normal Cave Spiders that spawn in the area poison often.

Spiderlings will attempt to run to Gargantula's position when summoned, so melee fighters should be especially careful to avoid a damage spike.

AoE skills are very useful during this fight, both to deal with the spiderlings quickly and to catch stray regular spiders that spawn occasionally.

Veteran Cave Spiders normally inhabit the cave where Gargantula is located. Be prepared to deal with them after the event is over.